-38=
the total degree of all these elewents is v + 1. Since this is true
for each of the factors of the product, when the logs of the factors
are added and the s's formed, every s the sum of whose indices is
less than v + 1 will be zero. (See formula (7).)
The weight of Syp,eep is o+ B + °** 4 1), & <P <Cavee <A,
If a =1, then the s of lowest weight appearing in the expansion of

the product, is s,,, and the expansion begins with 1.+ smz"“
el

} is the
sans as given on page 11 except that the term a, x* does not appear

If 0,(x)==an®, then log {[1 + va(x)] e

at the beginning., When the sum of the logs is taken, s,2% does not
appear. All the other s's appear as given by formulas (7) and (11).

A
oo il %, where

Similarly, if 0, (x)=
m=j+k+*+r~1 ad h=jo+ i+ **+r\, then sji...,r will be
zerc in the coefficients given by (7) and (11) and nc other changes
will be made. :

As an example of a product of this type, expand

=

F(LS = x X101 +§] e " , where Yy denotes Euler's constant, that
X

is, vy = 0.5772156°""

Ietcu:,Zolg, ot R
n

& For the infinite product

1%

II[1 + f 1 e ™ we have sy = 0, sy =04, i>1 ., Using equations

          

 

 

(12) and expanding e'*, we have
1 Y %> (,x Ogx®
= 4 — -.. ——. tep e
RERE i ettty o

4
"*[1+’+(L J)’ iR ‘YG,#&T,);-!(Y '6YG'2+E’(51 6‘7‘435!’)'4"','1'.

It is to be noticed that these coefficients are exactly the C's

with vy instead of sy, and 0; instead of s; if i>1 ,

 

27 For the nuperical values of these sums, see Chrystal, Vol.II, p. 867,